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\def \aa{aa}
\def\bb{bb}
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\title{\vskip -40pt           % move title up
\huge \bf The Forced Duffing Oscillator 
\footnote{This file is from the 3D-XplorMath project. You can find it on the web by searching the name.}
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\begin{document}
\LARGE

\maketitle

\vskip -90pt  


%\centerline{\includegraphics{Duffing.png}}
%\medskip
% \centerline{  \hglue -22pt Unforced Duffing Oscillator  \hskip 25pt  Forced Duffing Oscillator}


\section{\LARGE What is it?} \vskip -25pt
  What we shall call the {\it Forced Duffing Oscillator Equation\/} is  the second order ODE for
 a single variable $x$,
  \begin{equation}
   {d^2x\over dt^2} =  -\hh\, x - ii \, x^3 -  \aa\, { dx \over dt} + \bb\, \cos( \cc\,t), \label{FDOE}
   \end{equation}
 whose solutions we display via the equivalent (non-autonomous) 
 first order system in two variables, $x$ and $y$: \hfil\break
 \centerline{
  \hfill$ {dx\over dt}= y, \ \ \ {dy\over dt} =  -\hh\, x - \ii \, x^3 -  \aa\, y + \bb\, \cos( \cc\,t) $ \hfill (2)
   }  
which in turn can be made into an autonomous first order system in three variables, $T$, $x$ and $y$: \hfil\break
\centerline{
\hfill $ {dT\over dt} = 1, \ {dx\over dt}  = y,\   {dy\over dt} =  -\hh\, x - \ii \, x^3 -  \aa\, y + \bb\, \cos( \cc\,T).$ \hfill (3)
   }
   We discuss the interpretation and significance of the five parameters, $\aa, \bb, \cc, \hh,\ii$ below. 
 Their default values are: $\aa = 0.25, \bb = 0.3, \cc = 1.0, \hh = -1.0,  \hbox{ and } \ii = 1.0$.   

  \section{\LARGE   Why is it interesting?} \vskip -25pt
   Here are two of the considerations that make the oscillator equation (\ref{FDOE}) worth studying.
First, with appropriate choices of parameter values it reduces to a variety of 
mathematically and physically interesting oscillator models; some classical such as the harmonic 
oscillator (with and without damping and forcing) and others that are more exotic, such as the 
classic Duffing oscillator introduced by Duffing in 1918. By putting these together in a parametric family, we can 
investigate how various features of these systems behave as we move around in the parameter space.
Secondly---and more importantly---it was in in the study of the Duffing Oscillator that symptoms of
the phenomena we now call ``chaos'' and  ``strange attractor'' were first glimpsed (although their 
significance was only appreciated later). By the Poincar\'e-Bendixson Theorem, three is the
smallest dimension in which an autonomous system can exhibit chaotic behavior, and the Duffing 
system is so simple that it lends itself very easily to the study and visualization of the phenomena 
related to chaos.

  \section{\LARGE   The Newtonian Particle Interpretation.} \vskip -25pt
  Note that (\ref{FDOE}) becomes  Newton's equation of motion for a particle of unit mass moving 
  on the $x$-axis if we define the ``force'', $F(x,{dx\over dt},t)$, acting on the particle to be the 
  right-hand side of (1). Let's interpret the various terms of $F$ from this point of view. 
  
  If $\hh$ is positive then the term $-\hh\,x$ by itself gives Hooke's Law for a spring, that ``stress is proportional to 
  strain'' and  the parameter $\hh$ has the interpretation of Hooke's proportionality factor between the 
  extension of the spring, $x$,  and the restoring force. 
   If also $\ii = 0$ then we have a pure Hooke's Law force that gives the Harmonic Oscillator, 
  ${d^2x\over dt^2} =  -\hh\, x $. But a real spring only 
  satisfies Hooke's Law approximately, and the term $-\ii \, x^3$ represents the next term in the 
  Taylor expansion of the restoring force under the reasonable assumption that this force is an 
  odd function of the spring extension, $x$. (If $\ii$ is positive it is called a ``hardening'' spring and if 
  negative a ``softening'' spring.)  For the classic Duffing Oscillator, $\hh$ is negative and $\ii$ is positive 
  and there is not a good interpretation of the force in terms of a spring. Rather, the sum of the two terms 
  $-\hh\, x - ii \, x^3$ should be interpreted as the force on a particle that is moving  in a double-well
   potential as we will discuss in more detail below.
  
  The term $-  \aa\, { dx \over dt}$ represents a ``friction'' force of the
  sort that would be experienced by a particle like a bullet traveling through air or a bead sliding on a 
  wire; that is, assuming that the ``damping'' or ``friction'' coefficient $\aa$ is positive, it describes a
  force acting on the particle in the direction opposite to the velocity and with a magnitude that is 
  proportional to the magnitude of the velocity. 
  
    Under the sum of the above terms of the force law $F$, the particle will (in general) oscillate 
 back and forth---which of course is why it is called an oscillator---however if $\aa > 0$ these
 oscillations will gradually die down as the kinetic energy is absorbed by friction. The final term 
 in the force law, $ \bb\, \cos( \cc\,t)$ is a periodic forcing term that will act on and perturb the motion of this 
 oscillating particle, and we note that it is solely a function of the time and is independent of both 
 the position and velocity of the particle. We will discuss a possible physical interpretation of this
term later.  The parameter $\bb$ is clearly the amplitude of this forcing term, i.e., its maximum magnitude, 
 and the parameter $\cc$ is the angular velocity of its phase in radians per unit time, so that the 
 period of the forcing term is $2 \pi \over \cc$ and its frequency is $\cc \over 2\pi$. As we shall see, it is 
 the energy that is fed into the system by this forcing term that is essential for the interesting chaos 
 related effects to occur. In fact the most interesting behaviors of solutions of (1) are present when all 
 the above terms are present in $F$, that is when the oscillator is both forced and damped, and in fact 
 the way damping and forcing can balance each other is crucial to understanding the general behavior of solutions. 
 However we will begin by analyzing the simpler situation when both the damping and forcing terms are missing.
  

  \section{\LARGE   The Undamped, Unforced Case.} \vskip -25pt
  We now assume that $\aa$ and $\bb$ are both zero, so the force $F(x) = -\hh \,x - \ii\, x^3$ is a function of
  $x$ alone. Now in one-dimension, whenever this is the case the force is {\it conservative\/}, 
  that is, it is minus the derivative of a ``potential'' function, $U(x)$. Indeed, if we define
  $U(x) := -\int_0^x F(\xi) \, d\xi$, then clearly $F(x) = -U'(x)$. 
  If as above we write $y := {dx\over dt}$, define the kinetic energy by 
  $K(y) := {1\over 2} y^2$ and define the Hamiltonian or total energy function  by $H(x,y) := K(y) + U(x)$,
  then ${d H \over dt} = y\, {dy\over dt} + U'(x){dx\over dt} = y({dy\over dt}  + U'(x)) $. So, if Newton's 
  Equation is satisfied, ${dy\over dt} = {d^2x\over dt^2} =  F(x) = -U'(x)$, so ${d H \over dt} = 0$.
  This of course is the law of conservation of energy:  the total energy function $H(x,y)$ is constant 
  along any solution of Newton's Equations. In one-dimension this provides at least in principle a way 
  to solve Newton's Equation for any initial conditions $x = x_0$ and $y = y_0$ at time $t = t_0$.
  Namely, the path or orbit of the solution is a curve in the $x$-$y$ plane, and by conservation of 
  energy this curve is given by the implicit equation $H(x,y) = H(x_0,y_0)$. And since 
  $\left({dx\over dt}\right)^2 = y^2 = 2 K(y) = 2(H(x_0,y_0) - U(x))$, we find: 
  $${dt\over dx} = {1\over \sqrt{2(H(x_0,y_0) - U(x))}},$$  so we can find $t$ as a function of $x$ 
  by a quadrature, and then invert this relation to find $x$ as a function of $t$.
  
    In the Harmonic Oscillator case, with $\hh = 1$ and $\ii = 0$, $U(x) = {1\over 2} x^2$ so 
  $H(x,y) = {1\over 2} (x^2 + y^2)$, so the orbits are circles, and it is easy to carry out the above 
  quadrature and inversion explicitly, to obtain $x(t) = x_0 \cos(t-t_0) + y_0 \sin(t-t_0)$.
  
   \section{\LARGE   The Universal ``Sliding Bead on a Wire'' Model.} \vskip -25pt
   In one-dimension there is a highly intuitive physical model that makes it easy to visualize the motion
of a particle under a given force. Moreover this model is ``universal'' in the sense that it works for all  forces 
that are  function of position only and hence, as we noted above, are of the form $F(x) = - U'(x)$ 
for some potential function $U$.  Namely, imagine that we string a bead on a frictionless 
piece of wire that lies along the graph of the equation $y = U(x)$. If the bead has mass $m = 1$ and if we 
choose units so that $g$, the acceleration of gravity, equals one, then the gravitational potential of the bead
is $mgh = h$ where $h$ is its height. So if as usual we interpret the ordinate of a point as its height, then 
the gravitational potential of the bead when it is at the point $(x,y) = (x,U(x))$ is just $U(x)$, and the
sliding motion of the bead along the wire under the attraction of gravity will exactly model whatever 
other system we started from!

  In the case of the Harmonic Oscillator, where $F(x) = - x$ and $U(x) = {1\over 2} x^2$, the graph is the 
parabola, $y = {1\over 2} x^2$ and it is easy to imagine the bead oscillating back and forth along this parabola.

For the unforced and undamped Duffing Oscillator the force is $F(x) = -\hh\, x - \ii \, x^3$, where for simplicity in what 
follows we will assume that $\ii > 0$ and $\hh < 0$. The potential is $U(x) = {\hh  \over 2} x^2  +  {\ii \over 4} x^4$, which we 
note can be considered as the first two terms in the Taylor expansion for an arbitrary symmetric potential with local 
maximum at $0$. It is easily checked that $\lim_{x \to \pm\infty} U(x) = +\infty$ and in addition to the local maximum 
at $0$, there are two other critical points of $U$, at $x = \pm \sqrt{-\hh\over \ii}$, where $U$ has local minima.
For the default values, $\hh = -1$ and $\ii = 1$, the force is $F(x) =  x( 1 - x^2)$, and the potential is 
$U(x) = {1\over 4} x^2( x^2 - 2)$,  so the local minima are at $\pm1$. We graph this force 
$F(x)$ and potential $U(x)$ below, and show a selection of the resulting orbits. It should be clear why $U$ is called 
a double-well potential.

\bigskip
\centerline{\includegraphics{ForcePotential.png}}
\medskip
\centerline{  \hglue -0.4 in $F(x)$  \hskip 2.6 in  $U(x)$}

\bigskip
\centerline{\includegraphics{Orbits.png}}
\medskip
\centerline{ Some Orbits of the Unforced, Undamped Duffing Oscillator}


  \section{\LARGE   The Unforced but Damped Duffing Oscillator.} \vskip -25pt
  
  We now still assume $\bb = 0$ (so there is no forcing) but we assume that $\aa > 0$, so there is damping.
In the bead on a wire picture, $\aa\,{dx\over dt} = \aa\, y$ is the friction from the bead rubbing against the wire, 
and the force is now given by $F(x) = -U'(x) - \aa\,y$. If we again calculate ${dH\over dt}$ as we did above, 
we now find not ${dH\over dt} = 0$ but instead ${dH\over dt} = -\aa \,y^2$. The result is that instead of the orbits 
of the bead in the $x$-$y$-plane being closed curves of constant total energy $H$, the energy decrease along the 
obits, and they cut across the $H =\ $ constant curves and spiral in towards the two minima of $H$ at the bottom
of the two potential wells.  We show a selection of the resulting orbits below. 

\bigskip
\centerline{\includegraphics{DampedOrbits.png}}
\medskip
\centerline{ Some Orbits of the Unforced, Damped Duffing Oscillator}

  \section{\LARGE   The Forced Duffing Oscillator.} \vskip -25pt
    We now add back the forcing term $ \bb\, \cos( \cc\,t)$. First a word about how to interpret this force in the sliding bead picture.
 If we assume that there is an alternating electric field parallel to the $x$ direction and with strength $\cos( \cc\,t)$ at
 time $t$, then $ \bb\, \cos( \cc\,t)$ will be the electric force felt by the bead if we give the bead an electric charge of 
 magnitude $\bb$.

\bigskip
\centerline{\includegraphics{FullDuffingOrbits.png}}
\medskip
\centerline{ Some orbits of the Forced, Damped Duffing Oscillator}

 \section{\LARGE   Chaos and Strange Attractors.} \vskip -25pt


\bigskip
\centerline{\includegraphics{DuffingAttractor2Slices.png}}
\medskip
\centerline{Two Slices of the Duffing Attractor}



\vskip 12pt

\author{R.S.P.}

		        
\vfil\eject 



\bye